Abstract
Abstract In this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of R n , n ≥ 4 . Under the assumption that the prescribed function is flat near its critical point, we give precise estimates on the losses of the compactness, and we provide a new existence result of Bahri-Coron type. Moreover, we establish, under generic boundary condition, a Morse inequality at infinity, which gives a lower bound on the number of solutions to the above problem. MSC:58E05, 35J65, 53C21, 35B40.
Highlights
In this paper we consider a nonlinear elliptic equation involving the Sobolev trace critical exponent associated to conformal deformations of Riemannian metrics on manifolds with boundary
Its boundary is denoted by Sn– and it is endowed with the standard metric still denoted by g
We study the problem of finding a conformal metric g = u n– g such that Rg = in belongs to H (Bn) and hg = H on Sn
Summary
In this paper we consider a nonlinear elliptic equation involving the Sobolev trace critical exponent associated to conformal deformations of Riemannian metrics on manifolds with boundary. Rg is the scalar curvature of the metric g in Bn and hg is the mean curvature of g on Sn– This problem has the following analytical formulation: find a smooth positive function which solves the following nonlinear boundary value equation:. If ind(H, y) denotes the Morse index of H at the critical point y, problem ). Namely we study the noncompact orbits of the gradient-flow of J, the so-called critical points at infinity following the terminology of Bahri [ ]. In Section , we set up the variational problem and we recall the expansion of the gradient of the associated Euler-Lagrange functional near infinity. If w = , it is called of w-type or mixed type With such a critical point at infinity, stable and unstable manifolds are associated.
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